Differential Equations and Numerical Analysis Seminar - 28/03/2012

Wednesday, 28 March 2012, 12:00 p.m.

Lecturer: Filip Rindler, University of Cambridge

Title: "Lower semicontinuity and Young measures in the space BD of functions of bounded deformation".

Local: Room 1.5, Edifício VII
Faculdade de Ciências e Tecnologia, Quinta da Torre, Caparica

Abstract: The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (Du + Du^T)/2 is representable as a finite Radon measure. Such functions play an essential role in modern theories of (linear) elasto-plasticity in a variational framework. In this talk, I will present results about the solvability of minimization problems for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. This includes the case of a non-vanishing Cantor-part in the symmetrized derivative, corresponding to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows to prove the crucial lower semicontinuity result without an Alberti-type theorem in BD, which is not available at present. A similar strategy also considerably simplifies the proof of the classical lower semicontinuity theorem by Müller & Fonseca and Ambrosio & Dal Maso in the space BV of functions of bounded variation.
Finally, I will present recent results about the classification of Young measures generated by sequences in BD.